| L(s) = 1 | + (76.8 − 133. i)3-s + (−1.04e3 − 1.80e3i)5-s + (489. + 6.33e3i)7-s + (−1.95e3 − 3.39e3i)9-s + (−2.80e3 + 4.85e3i)11-s + 1.15e5·13-s − 3.20e5·15-s + (−2.72e5 + 4.71e5i)17-s + (4.47e4 + 7.74e4i)19-s + (8.80e5 + 4.21e5i)21-s + (9.67e5 + 1.67e6i)23-s + (−1.19e6 + 2.07e6i)25-s + 2.42e6·27-s − 2.89e6·29-s + (7.67e5 − 1.32e6i)31-s + ⋯ |
| L(s) = 1 | + (0.547 − 0.948i)3-s + (−0.746 − 1.29i)5-s + (0.0770 + 0.997i)7-s + (−0.0994 − 0.172i)9-s + (−0.0576 + 0.0999i)11-s + 1.12·13-s − 1.63·15-s + (−0.790 + 1.36i)17-s + (0.0787 + 0.136i)19-s + (0.987 + 0.472i)21-s + (0.720 + 1.24i)23-s + (−0.613 + 1.06i)25-s + 0.877·27-s − 0.759·29-s + (0.149 − 0.258i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0137i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.068952352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.068952352\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-489. - 6.33e3i)T \) |
| good | 3 | \( 1 + (-76.8 + 133. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (1.04e3 + 1.80e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (2.80e3 - 4.85e3i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 - 1.15e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + (2.72e5 - 4.71e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-4.47e4 - 7.74e4i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-9.67e5 - 1.67e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + 2.89e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (-7.67e5 + 1.32e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-7.31e6 - 1.26e7i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 - 1.32e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 6.41e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + (2.47e7 + 4.29e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (3.02e7 - 5.24e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-6.25e7 + 1.08e8i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-5.29e7 - 9.17e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.16e8 + 2.01e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 1.09e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (-1.35e8 + 2.34e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (-1.39e8 - 2.42e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 - 2.55e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-5.80e8 - 1.00e9i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 - 1.67e8T + 7.60e17T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.09107351802395972395963361562, −11.11516864370182062669326366180, −9.237366964687435479646506073188, −8.421818496995733955495513079277, −7.88338987454850340946740862309, −6.34186806149982499127673373703, −5.04348979980558621375695905812, −3.65666474088041679522482905594, −1.97767891201822972921273410547, −1.08703921599467211063749840104,
0.57345095819528961045608844167, 2.79126644973992352138538784107, 3.67431706825193555242806750348, 4.53863929857695024858514970588, 6.56909812752189001527984454918, 7.40650221171325232945100459168, 8.701624646313833100371750293033, 9.844330842535507189726485444534, 10.93775009897511144595057497383, 11.23510917112311491906308253977
